The report states that the calculations were made using the automatic ‘pings’ sent to the satellite via the ground station and the aircraft after it vanished.
It explained that if the ground station does not hear from an aircraft for an hour it will transmit a ‘log on/log off’ message – a ‘ping’ – and the aircraft automatically returns a short message indicating that it is still logged on, a process described as a ‘handshake’.
The ground station log recorded six complete handshakes after ACARS, the aircraft’s operational communications system, stopped sending messages.Refined analysis
Inmarsat was then able to calculate the range of the aircraft from the satellite, and the time it took the signal to be sent and received, to generate two arcs of possible positions – a northern and a southern corridor.

OK, this is for the “arcs”.
This follows:

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The report goes on to explain that Inmarsat developed a second innovative technique that took into account the velocity of the aircraft relative to the satellite and the resulting change in signal frequency, known as the Doppler Effect.
The Inmarsat technique analysed the difference between the frequency that the ground station expected to receive and the one actually measured, known as the Burst Frequency Offset.